

A268216


Triangle read by rows: T(n,k) (n>=2, k=3..n+1) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,1,2,...,s,n} where s is the size of the largest proper open set in t.


7



2, 3, 6, 4, 12, 24, 5, 20, 60, 120, 6, 30, 120, 360, 720, 7, 42, 240, 840, 2520, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800
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OFFSET

2,1


LINKS

Table of n, a(n) for n=2..46.
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174179.


EXAMPLE

Triangle begins:
2,
3,6,
4,12,24,
5,20,60,120,
6,30,120,360,720,
7,42,240,840,2520,5040,
8,56,336,1680,6720,20160,40320,
9,72,504,3024,15120,60480,181440,362880,
10,90,720,5040,30240,151200,604800,1814400,3628800,
...


MATHEMATICA

i = 1; Table[Table[Binomial[n, i] FactorialPower[n  i, k], {k, 0,
n  i  1}], {n, 2, 9}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)


CROSSREFS

Row sums give A038156. A008279 (main entry).
Triangles in this series: A268216, A268217, A268221, A268222, A268223.
Cf. A282507.
Sequence in context: A097275 A130879 A119741 * A245712 A285331 A237125
Adjacent sequences: A268213 A268214 A268215 * A268217 A268218 A268219


KEYWORD

nonn,tabl


AUTHOR

N. J. A. Sloane, Jan 29 2016


EXTENSIONS

Definition clarified by Geoffrey Critzer, Feb 19 2017


STATUS

approved



